44 research outputs found
Existence, unique continuation and symmetry of least energy nodal solutions to sublinear Neumann problems
We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c
l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 &
\textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where is a bounded domain, and . For ,
will be identified with \sgn(u). We establish a variational
principle for least energy nodal solutions, and we investigate their
qualitative properties. In particular, we show that they satisfy a unique
continuation property (their zero set is Lebesgue-negligible). Moreover, if
is radial, then least energy nodal solutions are foliated Schwarz
symmetric, and they are nonradial in case is a ball. The case
requires special treatment since the formally associated energy functional is
not differentiable, and many arguments have to be adjusted
The second eigenvalue of the fractional Laplacian
We consider the eigenvalue problem for the {\it fractional Laplacian} in
an open bounded, possibly disconnected set , under
homogeneous Dirichlet boundary conditions. After discussing some regularity
issues for eigenfuctions, we show that the second eigenvalue
is well-defined, and we characterize it by means of several
equivalent variational formulations. In particular, we extend the mountain pass
characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and
nonlinear setting. Finally, we consider the minimization problem We prove that, differently from the
local case, an optimal shape does not exist, even among disconnected sets. A
minimizing sequence is given by the union of two disjoint balls of volume
whose mutual distance tends to infinity.Comment: 38 pages. The test function used in the proof of Theorem 3.1 needed
to be slightly modified, in order to be admissible for . We fixed this
issu
Asymptotic behaviour of higher eigenfunctions of the p-Laplacian as p goes to 1
Subject of this thesis is the asymptotic behaviour of the higher eigenvalues of the p-Laplacian operator as p goes to 1. The limit setting depends only on the geometry of the domain. In the particular case of a planar disc, it is possible to show that the second eigenfunctions are nonradial if p is close enough to 1. Moreover, it is shown that second eigenfunctions can be obtained as limit of least energy nodal solutions of a p-superlinear problem
The fractional Cheeger problem
Given an open and bounded set , we consider the
problem of minimizing the ratio between the perimeter and the
dimensional Lebesgue measure among subsets of . This is the
nonlocal version of the well-known Cheeger problem. We prove various properties
of optimal sets for this problem, as well as some equivalent formulations. In
addition, the limiting behaviour of some nonlinear and nonlocal eigenvalue
problems is investigated, in relation with this optimization problem. The
presentation is as self-contained as possible.Comment: 33 pages, 2 figure